Corks, involutions, and Heegaard Floer homology

Dai, Irving; Hedden, Matthew; Mallick, Abhishek

doi:10.48550/arxiv.2002.02326

Cited by 6 publications

(31 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The results of [DHM20] easily imply that V 0 and V 0 are invariant under equivariant concordance (essentially by surgering along the concordance annulus). Hence it is actually immediate that V 0 and V 0 obstruct equivariant sliceness.…”

Section: Introductionmentioning

confidence: 92%

“…This is in marked contrast to the usual approach taken in the literature, where in order to deploy various invariants, one (naturally) has in mind a specific family of slice disks (or surfaces) that are conjectured to be non-isotopic. The situation here is analogous to the notion of a strong cork introduced by Lin-Ruberman-Saveliev in [LRS17] and studied in [DHM20].…”

Section: Introductionmentioning

confidence: 93%

“…Instead of working with numerical invariants, it is also possible to define a local equivalence group in the style of Hendricks-Manolescu-Zemke [HMZ18] or Zemke [Zem19]. This follows the approach taken in Dai-Hedden-Mallick in [DHM20] to study cork involutions; and, indeed, the current article is closely related to [DHM20]. In this paper, we define the local equivalence group K τ,ι of pτ K , ι K q-complexes and show that there is a homomorphism from the equivariant concordance group r C (defined by Sakuma in [Sak86]) to K τ,ι :…”

Section: Introductionmentioning

confidence: 99%

“…Note that τ H and ι H do not in general commute. This is in contrast to the 3-manifold setting; see [DHM20,Lemma 4.4].…”

Section: Introductionmentioning

confidence: 99%

“…In [Mal22], the second author established a large surgery formula relating the action of τ K to the corresponding Heegaard Floer action of the 3-manifold involution. This latter action was defined and studied in [DHM20] in the context of the theory of corks. It follows immediately from the large surgery formula that (with appropriate normalization) the invariants V 0 and V 0 are none other than the numerical involutive correction terms referenced in [DHM20, Remark 4.5].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Equivariant knots and knot Floer homology

Dai¹,

Mallick²,

Stoffregen³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly nonequivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden, and provide a re-proof of a result due to Miller and Powell regarding stabilization distance. Our formalism suggests a possible route towards establishing the non-commutativity of the equivariant concordance group.

show abstract

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

“…Note that τ H and ι H do not in general commute. This is in contrast to the 3-manifold setting; see [DHM20,Lemma 4.4].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Equivariant knots and knot Floer homology

Dai¹,

Mallick²,

Stoffregen³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

Miller

Powell

2023

Journal of London Math Soc

View full text Add to dashboard Cite

We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let K$K$ be a strongly invertible genus one slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum #nK$\#^n K$ is at least n/4$n/4$. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.

show abstract

Equivariant Seiberg-Witten-Floer cohomology

Baraglia¹,

Hekmati²

2021

Preprint

View full text Add to dashboard Cite

We develop an equivariant version of Seiberg-Witten-Floer cohomology for finite group actions on rational homology 3-spheres. Our construction is based on an equivariant version of the Seiberg-Witten-Floer stable homotopy type, as constructed by Manolescu. We use these equivariant cohomology groups to define a series of d-invariants d G,c (Y, s) which are indexed by the group cohomology of G. These invariants satisfy a Frøyshov-type inequality under equivariant cobordisms. Lastly we consider a variety of applications of these d-invariants: concordance invariants of knots via branched covers, obstructions to extending group actions over bounding 4-manifolds, Nielsen realisation problems for 4-manifolds with boundary and obstructions to equivariant embeddings of 3-manifolds in 4-manifolds. DAVID BARAGLIA AND PEDRAM HEKMATIHence we obtain a canonical isomorphismThis shows that the Seiberg-Witten-Floer cohomology HSW * (Y, s) does not depend on the choice of metric g. By working in an appropriately defined S 1 -equivariant Spanier-Whitehead category in which suspension by fractional amounts of C is allowed, Manolescu defined the Seiberg-Witten-Floer homotopy type of (Y, s) to beThis is independent of the choice of g by (1.1) and (1.2). Now suppose that a finite group G acts on Y by orientation preserving diffeomorphisms which preserve the isomorphism class of s. Let g be a G-invariant metric on Y . Lifting the action of G to the associated spinor bundle determines an S 1 extensionManolescu's construction of the stable homotopy type SW F (Y, s, g) can be carried out G s -equivariantly, so that SW F (Y, s, g) may be promoted to a G sequivariant stable homotopy type. This is analogous to the construction in [42] of the P in(2)-equivariant Seiberg-Witten-Floer stable homotopy type of (Y, s) where s is a spin-structure on Y . The main difference is that in our construction, the additional symmetries that comprise the group G s come from symmetries of Y rather than internal symmetries of the Seiberg-Witten equations.We define the G-equivariant Seiberg-Witten-Floer cohomology of (Y, s) to beThe right hand side is independent of the choice of metric g by much the same argument as in the S 1 -equivariant case. We make some remarks concerning this construction.(1) In this paper we have chosen to work throughout with cohomology instead of homology. This is simply a matter of preference and we could just as well work with Seiberg-Witten-Floer homology groups. (2) Instead of Borel equivariant cohomology, we could take co-Borel cohomology or Tate cohomology, which correspond to the different versions of Heegaard-Floer cohomology, [39, Corollary 1.2.4]. (3) In a similar fashion we can also define the G-equivariant Seiberg-Witten-Floer K-theory KSW * G (Y, s) = K * +2n(Y,s,g) Gs (SW F (Y, s, g)).More generally we could use any generalised equivariant cohomology theory in which the Thom isomorphism holds. (4) We have not attempted to construct a metric independent G s -equivariant stable homotopy type. To do this one would need to sp...

show abstract

Corks, involutions, and Heegaard Floer homology

Cited by 6 publications

References 27 publications

Equivariant knots and knot Floer homology

Equivariant knots and knot Floer homology

Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

Equivariant Seiberg-Witten-Floer cohomology

Contact Info

Product

Resources

About